\(\int \frac {b+a x^6}{\sqrt [3]{x+x^3} (d+c x^6)} \, dx\) [2214]

Optimal result

Integrand size = 26, antiderivative size = 164 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 c}-\frac {a \log \left (-x+\sqrt [3]{x+x^3}\right )}{2 c}+\frac {a \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )}{4 c}+\frac {(-b c+a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 c d} \]

[Out]

Unintegrable

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(4506\) vs. \(2(164)=328\).

Time = 4.88 (sec) , antiderivative size = 4506, normalized size of antiderivative = 27.48, number of steps used = 88, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2081, 6847, 6857, 245, 2181, 384, 524, 455, 57, 631, 210, 31} \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=-\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {\sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {\sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {\sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{7/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,\frac {\sqrt [3]{-1} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{4/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,\frac {\sqrt [3]{-1} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [9]{-1} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,\frac {\sqrt [3]{-1} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{8/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {(-1)^{2/3} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{5/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {(-1)^{2/3} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{2/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {(-1)^{2/3} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {\sqrt {3} a \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 c \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{7/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{4/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [9]{-1} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{8/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{5/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{8/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{5/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{2/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{9 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{7/9} (b c-a d) \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{4/9} (b c-a d) \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [9]{-1} (b c-a d) \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {3 a \sqrt [3]{x^2+1} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 c \sqrt [3]{x^3+x}}+\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{7/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{4/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [9]{-1} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{8/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{5/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} x^{2/3}-\sqrt [9]{d} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} x^{2/3}-\sqrt [9]{d} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}+\sqrt [9]{d} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}} \]

[In]

Int[(b + a*x^6)/((x + x^3)^(1/3)*(d + c*x^6)),x]

[Out]

-1/12*((b*c - a*d)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -((c^(1/3)*x^2)/d^(1/3))])/(c^(8/9
)*d^(10/9)*(x + x^3)^(1/3)) + ((-1)^(1/3)*(b*c - a*d)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2,
-((c^(1/3)*x^2)/d^(1/3))])/(12*c^(8/9)*d^(10/9)*(x + x^3)^(1/3)) - ((-1)^(2/3)*(b*c - a*d)*x^(5/3)*(1 + x^2)^
(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -((c^(1/3)*x^2)/d^(1/3))])/(12*c^(8/9)*d^(10/9)*(x + x^3)^(1/3)) + ((-1
)^(1/9)*(b*c - a*d)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, ((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3)]
)/(12*c^(8/9)*d^(10/9)*(x + x^3)^(1/3)) - ((-1)^(4/9)*(b*c - a*d)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1
, 5/3, -x^2, ((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10/9)*(x + x^3)^(1/3)) + ((-1)^(7/9)*(b*c - a*d
)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, ((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(1
0/9)*(x + x^3)^(1/3)) - ((-1)^(2/9)*(b*c - a*d)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -(((-
1)^(2/3)*c^(1/3)*x^2)/d^(1/3))])/(12*c^(8/9)*d^(10/9)*(x + x^3)^(1/3)) + ((-1)^(5/9)*(b*c - a*d)*x^(5/3)*(1 +
x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -(((-1)^(2/3)*c^(1/3)*x^2)/d^(1/3))])/(12*c^(8/9)*d^(10/9)*(x + x^
3)^(1/3)) - ((-1)^(8/9)*(b*c - a*d)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -(((-1)^(2/3)*c^(
1/3)*x^2)/d^(1/3))])/(12*c^(8/9)*d^(10/9)*(x + x^3)^(1/3)) + (Sqrt[3]*a*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2
*x^(2/3))/(1 + x^2)^(1/3))/Sqrt[3]])/(2*c*(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 -
(2*(c^(1/3) - d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*c*(c^(1/3) - d^(1/3))^(1
/3)*d^(8/9)*(x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*((-1)^(1/3)*c^(1/3) + d^(1/
3))^(1/3)*x^(2/3))/(d^(1/9)*(1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*c*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*d^(8
/9)*(x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(
1/3)*x^(2/3))/(d^(1/9)*(1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*c*(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/3)*d^(8/9
)*(x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*c^(1/9)*(1 + x^2)^(1/3))/(c^(1/3) - d
^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) - ((-1)^(1/3)*(b*c -
a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*c^(1/9)*(1 + x^2)^(1/3))/(c^(1/3) - d^(1/3))^(1/3))/Sqrt[3]])/(6*S
qrt[3]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((-1)^(2/3)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*
ArcTan[(1 + (2*c^(1/9)*(1 + x^2)^(1/3))/(c^(1/3) - d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - d^(1/3))^
(1/3)*d^(8/9)*(x + x^3)^(1/3)) - ((-1)^(1/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*c^(1/9)*(1 + x
^2)^(1/3))/(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d
^(8/9)*(x + x^3)^(1/3)) + ((-1)^(4/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*c^(1/9)*(1 + x^2)^(1/
3))/(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*
(x + x^3)^(1/3)) - ((-1)^(7/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*c^(1/9)*(1 + x^2)^(1/3))/(c^
(1/3) + (-1)^(1/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^
3)^(1/3)) + ((-1)^(2/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*c^(1/9)*(1 + x^2)^(1/3))/(c^(1/3) -
(-1)^(2/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3
)) - ((-1)^(5/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*c^(1/9)*(1 + x^2)^(1/3))/(c^(1/3) - (-1)^(
2/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((
-1)^(8/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*c^(1/9)*(1 + x^2)^(1/3))/(c^(1/3) - (-1)^(2/3)*d^
(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((b*c - a
*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - c^(1/3)*x^2])/(12*c*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/
3)*d^(8/9)*(x + x^3)^(1/3)) - ((-1)^(2/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - c^(1
/3)*x^2])/(36*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((-1)^(5/9)*(b*c - a*d)*x^(1/3
)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - c^(1/3)*x^2])/(36*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)
*(x + x^3)^(1/3)) - ((-1)^(8/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - c^(1/3)*x^2])/
(36*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log
[d^(1/3) + c^(1/3)*x^2])/(9*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((-1)^(1/3)*(b*c - a*d)*x^(
1/3)*(1 + x^2)^(1/3)*Log[d^(1/3) + c^(1/3)*x^2])/(36*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) - ((
-1)^(2/3)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[d^(1/3) + c^(1/3)*x^2])/(36*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8
/9)*(x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[d^(1/3) + (-1)^(2/3)*c^(1/3)*x^2])/(12*c*(-((-
1)^(2/3)*c^(1/3)) + d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((-1)^(1/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*
Log[d^(1/3) + (-1)^(2/3)*c^(1/3)*x^2])/(36*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) - (
(-1)^(4/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[d^(1/3) + (-1)^(2/3)*c^(1/3)*x^2])/(36*c*(c^(1/3) + (-1)^(1
/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((-1)^(7/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[d^(1/3) + (-
1)^(2/3)*c^(1/3)*x^2])/(36*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) - (3*a*x^(1/3)*(1 +
x^2)^(1/3)*Log[x^(2/3) - (1 + x^2)^(1/3)])/(4*c*(x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[(
c^(1/3) - d^(1/3))^(1/3) - c^(1/9)*(1 + x^2)^(1/3)])/(12*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3))
- ((-1)^(1/3)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[(c^(1/3) - d^(1/3))^(1/3) - c^(1/9)*(1 + x^2)^(1/3)])/(1
2*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((-1)^(2/3)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[(
c^(1/3) - d^(1/3))^(1/3) - c^(1/9)*(1 + x^2)^(1/3)])/(12*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3))
- ((-1)^(1/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3) - c^(1/9)*(1 + x^2)
^(1/3)])/(12*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((-1)^(4/9)*(b*c - a*d)*x^(1/3)
*(1 + x^2)^(1/3)*Log[(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3) - c^(1/9)*(1 + x^2)^(1/3)])/(12*c*(c^(1/3) + (-1)^(1
/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) - ((-1)^(7/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[(c^(1/3) + (
-1)^(1/3)*d^(1/3))^(1/3) - c^(1/9)*(1 + x^2)^(1/3)])/(12*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x
^3)^(1/3)) + ((-1)^(2/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3) - c^(1/9
)*(1 + x^2)^(1/3)])/(12*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) - ((-1)^(5/9)*(b*c - a
*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3) - c^(1/9)*(1 + x^2)^(1/3)])/(12*c*(c^(1/3
) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((-1)^(8/9)*(b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[(
c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3) - c^(1/9)*(1 + x^2)^(1/3)])/(12*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(
8/9)*(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*x^(2/3)
- d^(1/9)*(1 + x^2)^(1/3)])/(4*c*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) - ((b*c - a*d)*
x^(1/3)*(1 + x^2)^(1/3)*Log[(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/3)*x^(2/3) - d^(1/9)*(1 + x^2)^(1/3)])/(4*c*(
-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Log[(c^
(1/3) - d^(1/3))^(1/3)*x^(2/3) + d^(1/9)*(1 + x^2)^(1/3)])/(4*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(x + x^3)^(1
/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 2181

Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c^3 + d^3*x
^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ
[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {b+a x^6}{\sqrt [3]{x} \sqrt [3]{1+x^2} \left (d+c x^6\right )} \, dx}{\sqrt [3]{x+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {b+a x^9}{\sqrt [3]{1+x^3} \left (d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {a}{c \sqrt [3]{1+x^3}}+\frac {b c-a d}{c \sqrt [3]{1+x^3} \left (d+c x^9\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}} \\ & = \frac {\left (3 a \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x+x^3}}+\frac {\left (3 (b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x+x^3}} \\ & = \frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 c \sqrt [3]{x+x^3}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x+x^3}}+\frac {\left (3 (b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x+x^3}} \\ & = \frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 c \sqrt [3]{x+x^3}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.03 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.17 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=-\frac {\sqrt [3]{1+\frac {1}{x^2}} x \left (3 a d \left (i \left (i+\sqrt {3}\right ) \log \left (\sqrt {2-2 i \sqrt {3}}-2 i \sqrt [3]{1+\frac {1}{x^2}}\right )+\left (-1-i \sqrt {3}\right ) \log \left (\sqrt {2+2 i \sqrt {3}}+2 i \sqrt [3]{1+\frac {1}{x^2}}\right )+2 \log \left (-1+\sqrt [3]{1+\frac {1}{x^2}}\right )\right )+2 (b c-a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{1+\frac {1}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{12 c d \sqrt [3]{x+x^3}} \]

[In]

Integrate[(b + a*x^6)/((x + x^3)^(1/3)*(d + c*x^6)),x]

[Out]

-1/12*((1 + x^(-2))^(1/3)*x*(3*a*d*(I*(I + Sqrt[3])*Log[Sqrt[2 - (2*I)*Sqrt[3]] - (2*I)*(1 + x^(-2))^(1/3)] +
(-1 - I*Sqrt[3])*Log[Sqrt[2 + (2*I)*Sqrt[3]] + (2*I)*(1 + x^(-2))^(1/3)] + 2*Log[-1 + (1 + x^(-2))^(1/3)]) + 2
*(b*c - a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , Log[(1 + x^(-2))^(1/3) - #1]/#1 & ]))/(c*d*(x +
x^3)^(1/3))

Maple [N/A] (verified)

Time = 0.24 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(\frac {\left (a d -b c \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 d \,\textit {\_Z}^{6}+3 d \,\textit {\_Z}^{3}+c -d \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )-3 \left (\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )-\frac {\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}\right ) d a}{6 d c}\)	\(148\)

[In]

int((a*x^6+b)/(x^3+x)^(1/3)/(c*x^6+d),x,method=_RETURNVERBOSE)

[Out]

1/6*((a*d-b*c)*sum(ln((-_R*x+(x*(x^2+1))^(1/3))/x)/_R,_R=RootOf(_Z^9*d-3*_Z^6*d+3*_Z^3*d+c-d))-3*(3^(1/2)*arct
an(1/3*(2*(x*(x^2+1))^(1/3)+x)*3^(1/2)/x)+ln(((x*(x^2+1))^(1/3)-x)/x)-1/2*ln(((x*(x^2+1))^(2/3)+(x*(x^2+1))^(1
/3)*x+x^2)/x^2))*d*a)/d/c

Fricas [F(-2)]

Exception generated. \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a*x^6+b)/(x^3+x)^(1/3)/(c*x^6+d),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

Sympy [N/A]

Not integrable

Time = 24.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.15 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int \frac {a x^{6} + b}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (c x^{6} + d\right )}\, dx \]

[In]

integrate((a*x**6+b)/(x**3+x)**(1/3)/(c*x**6+d),x)

[Out]

Integral((a*x**6 + b)/((x*(x**2 + 1))**(1/3)*(c*x**6 + d)), x)

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int { \frac {a x^{6} + b}{{\left (c x^{6} + d\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((a*x^6+b)/(x^3+x)^(1/3)/(c*x^6+d),x, algorithm="maxima")

[Out]

integrate((a*x^6 + b)/((c*x^6 + d)*(x^3 + x)^(1/3)), x)

Giac [N/A]

Not integrable

Time = 3.06 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int { \frac {a x^{6} + b}{{\left (c x^{6} + d\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((a*x^6+b)/(x^3+x)^(1/3)/(c*x^6+d),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int \frac {a\,x^6+b}{\left (c\,x^6+d\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]

[In]

int((b + a*x^6)/((d + c*x^6)*(x + x^3)^(1/3)),x)

[Out]

int((b + a*x^6)/((d + c*x^6)*(x + x^3)^(1/3)), x)